Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}4x+8y &= -8 \\ -4x+y &= -5\end{align*}$
Answer: We can eliminate $x$ when its corresponding coefficients are negative inverses. Add the top and bottom equations. $9y = -13$ Divide both sides by $9$ and reduce as necessary. $y = -\dfrac{13}{9}$ Substitute $-\dfrac{13}{9}$ for $y$ in the top equation. $4x+8( -\dfrac{13}{9}) = -8$ $4x-\dfrac{104}{9} = -8$ $4x = \dfrac{32}{9}$ $x = \dfrac{8}{9}$ The solution is $\enspace x = \dfrac{8}{9}, \enspace y = -\dfrac{13}{9}$.